Counting monic irreducible polynomials $P$ in ${\mathbb{F}_q[X]}$ for which order of ${X\!\!\hspace{4.44443pt}(\@mod \; P)}$ is odd

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Christian BALLOT

Counting monic irreducible polynomials P in Fq[X] for which order of

X (mod P ) is odd

Tome 19, no1 (2007), p. 41-58.

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de Bordeaux 19 (2007), 41-58

Counting monic irreducible polynomials P in

F

q

[X] for which order of X (mod P ) is odd

par Christian BALLOT

Résumé. Hasse démontra que les nombres premiers p pour les-quels l’ordre de 2 modulo p est impair ont une densité de Diri-chlet égale à 7/24-ième. Dans cet article, nous parvenons à imiter la méthode de Hasse afin d’obtenir la densité de Dirichlet δq de

l’ensemble des polynômes irréductibles et unitaires P de l’anneau Fq[X] pour lesquels l’ordre de X (mod P ) est impair. Puis nous

présentons une seconde preuve, nouvelle, élémentaire et effective de ces densités. D’autres observations sont faites et des moyennes de densités sont calculées, notamment la moyenne des δp lorsque

p parcourt l’ensemble des nombres premiers.

Abstract. Hasse showed the existence and computed the Dirich-let density of the set of primes p for which the order of 2 (mod p) is odd; it is 7/24. Here we mimic successfully Hasse’s method to compute the density δq of monic irreducibles P in Fq[X] for which

the order of X (mod P ) is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the δp’s as p varies through all rational primes.

1. Introduction

In a 1958 paper [Sier], Sierpinski considered the partition of the rational primes p into two sets: the primes for which the order of 2 (mod p) is odd and those for which this order is even. He asked what the relative size of these two sets was. A satisfying and complete answer to this question was found by Hasse [Ha] in the mid-sixties: the odds in favour of the order of 2 (mod p) being odd versus even are 7 to 17. That is, in the sense of Dirichlet, or of natural density, the order of 2 is odd for 7 out of 24 primes. In this paper, we investigate a similar question but in a new setting. Given a prime power q we consider the euclidean ring Fq[X] and ask for the proportion

of primes P in Fq[X] for which the order of X (mod P ) is odd. Here a

prime P is a monic irreducible polynomial of Fq[X]. It will be seen that Manuscrit reçu le 26 juillet 2005.

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the answer depends heavily on q, but we will show among other results that the 7 to 17 odds are again the correct answer if and only if q is an odd power of a prime congruent to 3 (mod 8). For any other odd q’s, these odds are lower than 7 to 17.

Besides the introduction the paper contains five sections. Section 2 is pre-liminary and is divided into four subsections. Section 2.1 introduces nota-tion. Section 2.2 is devoted to brief remarks and definitions about Dirichlet and natural densities for primes in Z or in Fq[X]. Section 2.3 contains the

statements of three relevant analogues of classical theorems: the Kummer-Dedekind prime decomposition theorem, the Dirichlet and Kronecker Den-sity Theorems. In Section 2.4 a technical proposition about the power of 2 dividing qn− 1 is proven in a most elementary way. This result will be of frequent use.

The calculation à la Hasse of the density δq of primes P in Fq[X] such

that X has odd order modulo P is carried out in Section 3. The method devised by Hasse to compute the density of rational primes p for which the order of a (mod p) is odd (where a is any integer, not just 2) proceeded by first calculating the sub-densities of such primes within the primes p having a fixed power of 2 dividing p − 1. The method we employ here works similarly with sub-densities being computed within the primes P having a fixed power of 2 dividing their degrees. We will indicate why our method constitutes a proper generalization of the classical method of Hasse (see Remark 3.4).

Now the sub-density corresponding to primes of odd degree (i.e. primes whose degree is divisible by 20) obtained in Section 3 yields the exact pro-portion of primes P , within primes of degree n ≥ 3, for each odd integer n, for which the order of X (mod P ) is odd. This, we prove in Section 4 in an elementary manner. This observation and the idea, used to mimic the Hasse method in Section 3, of separating primes according to the 2-adic valuation of their degree, are put together to yield another proof of the existence and of the values of the densities computed in Section 3. This new proof, however, is elementary in that it does not use the Kummer-Dedekind, nor the Kronecker Density Theorems; the proof actually shows that our sets of primes have a natural density, via a notion of natural density that was shown to imply Dirichlet density in [Ba2]. Moreover, the proof is effective in that it shows how the natural density is being approached when considering only primes up to a given norm qN. Section 4 also contains additional re-marks, one of which, Remark 4.6, is of computational interest and presents numerical data for q = 3 and primes P of small degrees.

Section 5 contains two subsections. In the first one we return to the classical case to calculate, through a dual and average point of view, and given a “typical" integer a, the expected density of primes p such that a

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has odd order (mod p). Indeed the 7:17 odds found by Hasse for a = 2 is not typical. These odds are 1:2 for most integers a and in particular for a equal to any odd prime q. Loosely speaking we may say that having chosen a “random" pair of primes (q, p), the odds that the order of q (mod p) be odd are 1 to 2. Thus, this statement made in “dual" form leads us to a short heuristic argument, less educated than the proof of Hasse, to account for the proven, yet still intriguing fact that the order of q (mod p) is twice as often even than odd. In Section 5.2, we carry this point of view over to Fq[X] and, as a consequence, we show that the choice of X rather than

that of another monic polynomial in Fq[X] is representative of the general

expected odds.

That the densities δq computed in Section 3 vary a great deal with q, and

that the point of view of Section 5 be successful, suggests we investigate the densities themselves “on average". In the classical setting this average is 1/3. What will the densities in our setting turn out to be on average? Thus, in Section 6, we define in reasonable manner density averages over δq’s. We do so either as q varies through the same eth power of primes p,

or as q varies among all powers of the same prime p, or simply over all prime powers q. We show how to prove their existence and we compute their values. For instance, we define δ(−, e = 1), the average of the δp’s

over all rational primes, as limx _π(x)1 Pp≤xδp. Various asymptotic averages

over rational primes have recently been investigated. For instance the quan-tity 1_xP

p≤x α(p−1)

p−1 , where α(p − 1) is the average order of elements in F ∗ p,

was studied in [Ga], or δ(a, d) = _π(x)1 P

p≤xδ(p; a, d), where δ(p; a, d) is the

proportion of elements of F∗p whose order is congruent to a (mod d), was

studied in [Mo2]. Note that taking d = 2 and a = 1 in the last example is akin to our problem. In fact, see Remark 4.5, finding δ(1, 2) is like averaging over rational p’s the proportions of P ’s for which the order of X (mod P ) is odd among primes P in Fp[X] of degree 1. Instead, we may say that our

δ(−, e = 1) is an average over rational p’s of the proportions of such P ’s regardless of degree.

Much work and progress has been done, since Sierpinski, to compute densities for rational primes defined in several ways that generalize the primes of the Sierpinski question. Thus, many densities for primes in Fq[X]

are expected to be computable.

For instance, finding the density of primes P in Fq[X] with order of X

(mod P ) even, as we did here, is the same as determining the density of primes dividing some term of the second order linear recurrence (Xn ₊

1)n≥0. The Hasse method, as in the classical case, clearly generalizes to

other special recurrences, such as the companion Chebyshev-like recurrence defined by Tn+2= XTn+1+ Tn, n ≥ 0 with T0 = 2 and T1= X in Fq[X], q

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odd. But would the elementary method of Section 4 also generalize to such a sequence Tn?

Also, if ` is any prime, not just 2, arguments analogous to those of Sec-tions 3 or 4, yield the density of primes P in Fq[X] with order of X (mod P )

divisible by ` (see [Ba3]).

Acknowledgment. I thank the anonymous Referee who read this paper

carefully and made sensible suggestions.

2. Preliminaries

2.1. Notation. We denote the exponent of the power of 2 dividing an

integer n by ν2(n). Equivalently, we will write 2ν2(n)||n or say that 2ν2(n)is

the (exact) power of 2 in n.

Having fixed a rational prime p and an integer e ≥ 1, we set q = pe. A prime P in Fq[X] is a monic irreducible polynomial of the ring. The

degree of a polynomial P is written deg(P ) and the size of the quotient ring Fq[X]/P , i.e. the norm of P , is denoted by |P |. The field of fractions

of the ring Fq[X] is denoted by K.

The symbol ζn denotes a primitive n-th root of unity in the algebraic

closure Fp of Fp.

We define for all integers k ≥ 0 the sets Sq(k) to be

{P ∈ Fq[X], P is prime and 2k||deg(P )}.

2.2. Dirichlet and natural densities of sets of primes. Let S be a

set of rational primes. Provided the limits exist, the natural density of S is defined as d(S) = limN →∞N−1PNn=1

χS(pn)

1 , where pn is the n-th rational

prime and χSis the characteristic function of S, while the Dirichlet density

of S is δ(S) = lim_s→1+ P p∈Sp −s P pp −s .

It is well-known that a set S with a natural density d has a Dirichlet density δ = d ([Des], Chap. 8). However, a set of rational primes may have a Dirichlet density, but no natural density ([Ser], p. 76). The Chebotarev Density Theorem is usually stated with respect to Dirichlet density. How-ever all Chebotarev sets of primes, i.e. sets to which the Chebotarev Density Theorem or one of its corollaries may be applied, all have a natural density ([Nar], Theorem 7.10*). The definition of natural density presents a com-putational advantage compared to that of Dirichlet in the sense that the approximants N−1PN

n=1χS(pn) can be calculated using only a few primes

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Definitions. A set S of primes in Fq[X] is said to have Dirichlet density δ

if the limit below exists and lim s→1+ P P ∈S|P |−s P P |P |−s = δ.

For a set S of primes in Fq[X], we define natural density by

(2.1) d(S) = lim N →∞ 1 N N X n=1 Sn In ,

where Sn represents the number of primes in S of degree n and In is the

number of primes of degree n in Fq[X], if the limit exists.

The reasons for defining d(S) as in (2.1) are discussed abundantly in [Ba2] where (2.1) is compared to other definitions and shown to yield the best analogue to the classical notion. In particular, the relations between Dirichlet and natural densities are similar to that of the classical case. Also the N -th approximants rN = _N1 PNn=1 SInn can be computed to provide an

empirical check to a density computed theoretically. In fact we do this in Remark 4.6.

2.3. Some relevant theorems : statement and reference.

Proposition 2.1. (A weak function field version of a classical

Kummer-Dedekind Theorem) Let α be an algebraic integer over K with minimal polynomial M (Y ) ∈ (Fq[X])[Y ] (that is M (Y ) is monic with coefficients in

Fq[X]) and P be a prime in Fq[X]. Then there is a one-to-one

correspon-dence between prime ideal factors P of the ideal generated by P in the ring of integers Oα of K(α) and monic irreducible factors M of M , and the

correspondence is such that the degree of Oα/P over Fq[X]/P matches the

polynomial degree of M.

Proof. One may view this proposition as a particular instance of Proposi-tion 25 of [Lan] with the Dedekind ring A = Fq[X]. Theorem 2.2 (Dirichlet Density Theorem). Let B and M be two

polynomi-als in Fq[X] with deg(M ) ≥ 1. Then the set of primes {P ∈ Fq[X]; P ≡ B

(mod M )} has Dirichlet density δ = 1/Φ(M ), where Φ(M ) is the number of non-zero polynomials prime to M and of degree less than deg(M ).

Proof. See [Ro], Theorem 4.7.

Theorem 2.3 (Kronecker Density Theorem). Let L be a Galois

exten-sion of K of Galois group G. Then the set of primes in Fq[X] which split

completely in L has Dirichlet density 1/|G|.

Proof. This is a special case of the Chebotarev Density Theorem

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2.4. A useful lemma. Let p be an odd prime, e an integer ≥ 1 and q = pe. We consider the pair of Lucas sequences un= q

n₋₁

q−1 and vn= qn+ 1 (n ≥ 0)

with characteristic polynomial X2−(q +1)X +q. As the reader may readily check we have the addition formula 2um+n= umvn+ unvm, ∀m and n ≥ 0,

and its corollary, the so-called double-angle formula, u2n = unvn, ∀n ≥ 0.

In the next proposition, we write x ∼ y to mean that the powers of 2 dividing the integers x and y are the same.

Proposition 2.4. For all integers n ≥ 0 we have that i) u_2n+1 is odd, and ii) u2n∼ n(q + 1).

Proof. First we prove i) by induction on n. If n = 0, then u2n+1 = u1 = 1

is odd. Assuming u2n+1 is odd for some n ≥ 0 then u2n+3= (q + 1)u2n+2−

qu2n+1 ≡ −qu2n+1 ≡ 1 (mod 2) and we are done. We then prove ii) by

in-duction on ν2(n). And we prove the initial step ν2(n) = 0 itself by induction

on the odd integers n ≥ 1. For n = 1 we have u2n = u2 = q + 1 ∼ n(q + 1) =

q + 1. Assume u2n ∼ q + 1 for some odd n, then by the addition formula

we have 2u2n+4 = u2nv4+ u4v2n. Now for any integer k ≥ 0 the integer vk

is even and v2k = (q2)k+ 1 ≡ 1k+ 1 ≡ 2 (mod 4) satisfies v2k ∼ 2. Hence

u2nv4 ∼ 2(q + 1) while u4v2n = u2v2v2n ∼ 2(q + 1)v2 so the power of 2

in u4v2n is larger than the power of 2 in u2nv4. Therefore u2n+4 ∼ q + 1

and the case ν2(n) = 0 holds. So let us assume ii) is true for any n with

ν2(n) = k ≥ 0. Then if n is such that ν2(n) = k + 1, write n = 2m. We have

using the inductive hypothesis u2n = u4m= u2mv2m∼ 2u2m∼ 2m(q +1) ∼

n(q + 1).

We restate Proposition 2.4 as a lemma in a form that will be of direct use throughout the paper.

Lemma 2.5. Let q = pe, where p is an odd prime and n an integer ≥ 1. Then

ν2(qn− 1) =

ν2(q − 1), if n is odd,

ν2(q2− 1) + ν2(n) − 1, if n is even.

Notation. We will denote by m the arithmetic function of n ≥ 1 defined

by m(n) = ν2(qn− 1).

Remark 2.6. By Lemma 2.5, m(n) depends only on ν₂(n).

3. Computation of δ_q via a method analogous to Hasse’s

In the proofs of Theorems 3.1 and 3.2, whenever the argument of the function m is dropped, we mean m(2k).

Theorem 3.1. For all integers k ≥ 0 the sets S_q(k) have a Dirichlet density δ(Sq(k)) where

δ(Sq(k)) =

1 2k+1.

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Proof. Given the prime p, our reckonings take place within a fixed algebraic closure Fpof Fp. The sets Sq(k), k = 0, 1, 2, . . ., partition the set I of primes

of Fq[X]. By Remark 2.6, the function ν2(|P | − 1) = ν2(qdeg P − 1) of the

prime P depends only on k. In fact, for P ∈ Sq(k), ν2(|P | − 1) = m(2k).

And we have, for the ideal P , the equivalence (3.1)

P ∈ Sq(k) ⇐⇒ P splits completely in K(ζ2m), but not in K(ζ₂1+m).

Indeed, P ∈ Sq(k) if and only if 2m, but not 2m+1, divides the order of

the cyclic group Fq[X]/P ∗

, or −1 is a 2m−1-th power, but not a 2m-th power in that cyclic group, or equivalently Y2m−1 + 1 = 0 is solvable in Fq[X] (mod P ), but Y2

m

+ 1 = 0 is not solvable. So that, applying Prop. 2.1 yields (3.1). Now by the Kronecker Density Theorem

(3.2) δ(Sq(k)) = [K(ζ2m) : K]−1− [K(ζ₂1+m) : K]−1.

But the dimension [K(ζ2m) : K] is also the degree of the extension F_q(ζ₂m)

over Fq (see Lemma 3.1.10 p. 64 of [Sti]).

Now the largest primitive 2x-th root of unity, ζ2x in F

q2k corresponds to

x = m(2k). Indeed F∗_q2k has order q

2k

− 1. And ζ2m is a primitive element

of F_q2k/Fq since a proper subfield of this extension (if any), i.e. any F_q2`,

` < k, has m(2`) < m(2k) by Lemma 2.5. Therefore F_q2k = Fq(ζ₂m(2k ))

so that, by (3.2) and the remark that follows (3.2), we have δ(Sq(k)) =

2−k− 2−k−1= 2−k−1.

Definition. Define Oq(k) as {P ∈ Sq(k); order of X (mod P ) is odd} and

Oq as Sk≥0Oq(k). The complement of Oq(k) in Sq(k) is Eq(k) = {P ∈

Sq(k); order of X (mod P ) is even}.

Theorem 3.2. The sets O_q(k) have a Dirichlet density δ(Oq(k)) where

δ(Oq(k)) = 2−m(2 k_)−k−1 = ( 2−ν2(p−1)−1_, _{if e is odd and k = 0,} 2−ν2(p2−1)−ν2(e)−2k_, _otherwise.

Proof. Let P be a prime of degree n in Sq(k). Then to say that X has

odd order (mod P ) means that X|P |−12m ≡ 1 (mod P ), or that there is

a solution Y ∈ Fq[X] satisfying Y2

m

− X ≡ 0 (mod P ), or again by Prop. 2.1 that the ideal P splits completely in Lk = K(ζ2m, 2m

√

X), but not completely in Lk(ζ2m+1) since P ∈ S_q(k). Now we have [L_k : K] =

[Lk : K(ζ2m)] × [K(ζ₂m) : K] = 2m × [K(ζ₂m) : K], and by the proof of

Theorem 3.1 we conclude that [Lk: K] = 2m× 2k, and also that Lk(ζ2m+1)

is a quadratic extension of Lk. Hence, by the Kronecker Density

Theo-rem, we get δ(Oq(k)) = 2−1 × 2−m(2

k_)−k

. By Lemma 2.5, if k = 0 and e is odd, then m(2k) = ν2(qn− 1) = ν2(q − 1) = ν2(p − 1) and otherwise

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m(2k_{) = ν}

2(pne− 1) = ν2(p2− 1) + ν2(ne) − 1 = ν2(p2− 1) + k + ν2(e) − 1

so that δ(Oq(k)) has the announced density. Remark. Because Dirichlet densities are finitely additive we deduce that

Eq(k) possesses such a density, namely δ(Sq(k)) − δ(Oq(k)).

Finally we compute the densities δq.

Theorem 3.3. The Dirichlet density δq of the set Oq of primes P in Fq[X]

for which the order of X (mod P ) is odd exists and is δq=      1, if p = 2, 2−ν2(p−1)−1_{+ 3}−1_{· 2}−ν2(p2−1)_, _{if e is odd,} 3−1· 2−ν2(p2−1)−ν2(e)+2_, _{if e is even.}

Proof. If p = 2 then the cyclic group (Fq[X]/P )∗ is of odd order 2e deg P− 1.

So every element has odd order, in particular X. Since X has odd order (mod P ) for any prime P ∈ Fq[X], we have δq = 1. For odd p, we claim

that δqexists and is the infinite sumPk≥0δ(Oq(k)) (Dirichlet densities are

finitely additive and since P

k≥0δ(Oq(k)) converges, the lower density is

at least the sum of the series, but a similar reasoning holds for primes for which X has even order which is the complementary set). Thus for e odd, we have δq = 2−ν2(p−1)−1+Pk≥12−ν2(p

2_−1)−2k

and for e even we also get δq by summing a geometric series of common ratio 1/4. Remark. It does not take long to check that for p not 2, δ_q is at most

1 4 +

1 24 =

7

24 and that this upper bound is reached exactly for q an odd

power of a prime p ≡ 3 (mod 8) (as claimed in the introduction).

Remark 3.4. In the classical Hasse method, odd primes p are partitioned

into sets Pj, j = 1, 2, 3, . . ., where Pj = {p; ν2(p − 1) = j}. Then for a

fixed j ≥ 1, necessary and sufficient splitting conditions on p in Pj are established for the order of 2 (mod p) to be odd, yielding for each j, as in Theorem 3.2 for each k, a sub-density dj. To compute the density of Oq in

the ring Fq[X] we computed the sub-densities of Oq within sets of primes

P having a fixed power of 2 in their degree n. By Lemma 2.5, the power of 2 in qn− 1 is a simple linear (or nearly linear with slope 1) function of the power of 2 in the degree n. But qn− 1 = |P | − 1. Thus, our approach replaces the p − 1 of the Hasse method of the classical setting by |P | − 1, a rather close analogue.

4. An elementary calculation of the densities based on counting primes

For any integer n ≥ 1, let Fn be the number of elements of odd order in

the multiplicative group F∗qnand let Gnbe the number of primitive elements

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Fn elements of odd order in F∗qn form a subgroup of F∗_qn. We recall that

the number In of primes P ∈ Fq[X] of degree n is given by the formula

In = _n1 P_d _nµ(d)qn/d, where µ is the Moebius function (see for instance

[Ro], Chap. 2).

Lemma 4.1. Let α ∈ Fq of degree n over Fq and let P denote the minimal

polynomial of α over Fq. Then

α has odd order in the multiplicative group of the field Fq(α) if and only

if P ∈ Oq.

Proof. Since Fq(α) ' Fqn, α has odd order in F∗_qn if and only if α is a

root of XFn_{− 1 or equivalently P divides X}Fn_{− 1 in F}

q[X], that is to say

P ∈ Oq.

Lemma 4.2. For any prime power q and any integer n ≥ 2, we have the

double inequality

qn− q q − 1q

n/2 _{< nI}

n< qn.

Proof. Not every element of Fqn is a root of an irreducible polynomial of

degree n, therefore nIn< |Fqn| = qn. Also

qn− nI_n= X 1<d n µ(d)qn/d ≤ n/2 X k=1 qk, since n d ≤ n 2, < q q − 1q

n/2_{, implying the other inequality.}

Theorem 4.3. Assume q = pe, p an odd prime and e ≥ 1. Let Ωn be the

number of primes in Oq of degree n. Then the ratio Ωn/In satisfies

( _Ω_n In = 2 −m(n)_, _{if n is odd ≥ 3,} 2−m(n)− _2(q−1)q q−n/2< Ωn In < 2 −m(n)₊ q 8(q−2)q −n/2_, _{if n is even,}

Proof. Since an element of a cyclic group of order x has odd order if and only if it belongs to the unique subgroup of order 2−ν2(x) _{· x, we}

have Fn = 2−ν2(q

n₋₁₎

· (qn_{− 1) = 2}−m(n)_(qn _{− 1). To compute G} n

ob-serve that Fn = P_d _nGd. So by the Moebius inversion formula we have

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Assume first that n is odd ≥ 3. Then any divisor d of n is odd and therefore m(d) = m(1) = m(n) by Remark 2.6. Hence

Gn= 2−m(n) X d n µ(d)(qn/d− 1) = 2−m(n)X d n µ(d)qn/d− 2−m(n)X d n µ(d), (4.1)

yielding Gn = 2−m(n)P_d _nµ(d)qn/d, since n > 1 implies P_d _nµ(d) = 0.

And Ωn= Gn/n = 2−m(n)In implies Ωn/In= 2−m(n).

Assume now that n is even. Then Gn= X d n µ(d)Fn/d= X d n µ(d)2−m(n/d)qn/d−X d n µ(d)2−m(n/d) ≥ 2−m(n)qn− 2−m(n/2)qn/2+ X d n, d≥3 µ(d)2−m(n/d)qn/d− 1/2, since 0 ≤P d nµ(d)2−m(n/d)≤ 1/2. Indeed, X d n µ(d)2−m(n/d)= X d 2l+1 µ(d)2−m(1)− X d 2l+1 µ(d)2−m(2) = ( 0, if l ≥ 1, 2−m(1)− 2−m(2)_, _{if l = 0,} where 2l + 1 = 2−ν2(n)_{n. But 0 < 2}−m(1)_{− 2}−m(2) _{< 2}−m(1) _{≤ 1/2.}

Proceeding as in Lemma 4.2, we get for n even ξ(n)=. X d n, d≥3 µ(d)2−m(n/d)qn/d ≤ 1 2(q − 1)q n/2₋1 2.

One can check directly the above inequality for n = 2 and 4, whereas for n ≥ 6 observe that ξ(n) ≤ 1 2 n/3 X k=1 qk ≤ q n/3+1_{− q} 2(q − 1) ≤ qn/2 2(q − 1) − 1 2. Hence, using −2−m(n/2)≥ −1/2, we have

Gn≥ 2−m(n)qn− 2−m(n/2)qn/2− _qn/2 2(q − 1) − 1 2 −1 2 ≥ 2−m(n)qn− q 2(q − 1)q n/2_.

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On the other hand, Gn ≤ 2−m(n)qn− ζ(n), where ζ(n) = 2. −m(n/2)qn/2 − P d n, d≥3µ(d)2 −m(n/d)_qn/d_{. But} ζ(n) > 2−m(n/2)qn/2− X d n, d≥6 µ(d)2−m(n/d)qn/d ≥ 2−m(n/2)qn/2− 2−m(n/2) n/6 X k=1 qk, which is > 0, since qn/2 >Pn/6

k=1qk for n ≥ 6, and where, in the above, we used µ(3) =

µ(5) < 0, µ(d) = 0, if 4 d, and 2−m(n/2) > 2−m(n). Therefore Gn <

2−m(n)qn.

Using the previous inequalities on Gn and Lemma 4.2 we get

2−m(n)qn−_2(q−1)q qn/2 qn < Ωn In = Gn nIn < 2 −m(n)_qn qn₋ q q−1qn/2 , or 2−m(n)− q 2(q − 1)q −n/2 < Ωn In < 2−m(n) 1 − q q − 1q −n/2 −1 < 2−m(n) 1 + q q − 2q −n/2_,

where the rightmost inequality above is obtained by writing 1/(1 − x) = 1 + x + x · x/(1 − x) with x = _q−1q q−n/2 and having _1−xx ≤ _{1−1/(q−1)}1/(q−1) = _q−21 , since qn/2 _{≥ q. We get the announced estimate by noting that n even}

implies m(n) ≥ 3, so that Ωn/In< 2−m(n)+_8(q−2)q q−n/2.

We state a fact mentioned in our introduction as a corollary of Theo-rem 4.3.

Corollary 4.4. The relative density ratio δ(Oq(k))/δ(Sq(k)) when k = 0 is

more than a density since it is the exact proportion of primes in Oq within

the set of primes of a given odd degree n ≥ 3, i.e. δ(Oq(0))/δ(Sq(0)) =

Ωn/In, for any odd n ≥ 3.

Proof. By Theorem 4.3 the ratio Ωn

In = 2

−m(n) _{which by looking at}

Theo-rems 3.1 and 3.2 is the ratio δ(Oq(k = 0))/δ(Sq(k = 0)). Remark 4.5. For n = 1, P

d nµ(d) = 1, so by (3.1) we get G1 =

2−m(1)(I1−1) = 2−m(1)(q − 1), instead of 2−m(1)q. That we get q − 1 instead

of q comes from the fact that there are only q − 1 primes of degree 1 for which the order of X (mod P ) is defined, namely P = X − a, for a ∈ F∗q.

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primes. Indeed, X ≡ a (mod X − a) implies the order of X (mod X − a) is the order of a in F∗q. This order is odd if and only if a belongs to the

subgroup of F∗q of order 2−ν2(q−1)· (q − 1) = 2−m(1)(q − 1) (a trivial case of

Lemma 4.1). Thus, Ω1/I1= 2−m(1)(q − 1)/q = 2−m(1)− q−12−m(1).

Remark 4.6. The numerical table below gives for q = 3 and each degree

n from 1 to 11 the values of In, Ωn and rn, where Ωn is the number of

primes in O3 of degree n and rn is the n-th approximant to d3 = δ3. See

§1.2 for the definition of the rn’s. They are given here with 4 decimal places

of accuracy.

n 1 2 3 4 5 6 7 8 9 10 11

In 3 3 8 18 48 116 312 810 2184 5880 16104

Ωn 1 0 4 1 24 13 156 25 1092 726 8052

rn .3333 .1666 .2777 .2222 .2777 .2506 .2859 .2540 .2813 .2655 .2868

The Ωn’s can be obtained by asking some mathematical software to factor

Xt− 1 (mod 3) for various t’s. But they can be computed directly by observing that if X has odd order modulo a prime P of degree n, then P must be a factor of some Xm− 1, where m is a factor of F_n, the largest odd factor in qn− 1. One only considers those factors m for which the order of q (mod m) is n. For in that case, the m-th cyclotomic polynomial Φm(X)

factors into ϕ(m)/n primes in Ωn, where ϕ is Euler’s totient function. As

an example consider q = 3 and n = 8. Then 38 − 1 = 25_{× 5 × 41. So}

F8 is 5 × 41. Since 3 has order 8 modulo 5 × 41 and modulo 41, there are

(4 · 40)/8 + 40/8 = 25 primes of degree 8 for which X has odd order. It is of interest to see how the successive rn’s approach the asymptotic

density δ3 = 7/24 which is about 0.2917.

Using Theorem 4.3 and the remarks about natural and Dirichlet densi-ties made in §2.2, we give a purely elementary proof of Theorem 3.3. By “elementary", we mean not using the Kummer-Dedekind Theorem, nor the Kronecker Density Theorem. This is a major difference from the classical case. In fact, the method we follow yields an effective theorem in which we precise how the sequence of approximants (rn) converges to the natural

density dq.

Theorem 4.7. For any N ≥ 1, we have, uniformly for any odd prime

power q = pe, e ≥ 1, r_N− δ_q < 3 4N,

where rN is the N -th approximant N−1PNn=1Ωn/In, δq is the Dirichlet

densityP

k≥02−k−1−m(2

k₎

of the set Oq found in Theorems 3.2 and 3.3. In

particular, Oq has a natural density dq= limNrN equal to δq.

Proof. We show that the set of primes Oq possesses a natural density dq =

P

k≥02−k−12−m(2

k₎

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dq. Since dq = limNN−1PNn=1Ωn/In, if the limit exists, we consider a sum

SN = PNn=1Ωn/In. Because Ωn/In depends essentially on k = ν2(n), we

need to estimate the number of integers n between 1 and N of the form 2k(2l +1) for a fixed k ≥ 0. Solving for l the inequality 2k(2l +1) ≤ N yields 0 ≤ l ≤ N −2₂k+1k. Thus there are

N +2k 2k+1

integers n with ν2(n) = k between 1

and N . But using Theorem 4.3 which, in particular, for n even implies that |Ωn/In− 2−m(n)| < .5q(q − 1)−1q−n/2 and the end of Remark 4.5, we get

(4.2) SN = X k≥0 _{N + 2}k 2k+1 2−m(2k)+ CN, where |CN| ≤ .5q(q − 1)−1Pn even ≤Nq−n/2− 2−m(1)q−1. Note that −1₂ ≤ ₂k+1N − N +2k 2k+1

< 1₂ so that replacing the integer value expressions

N +2k

2k+1

in (4.2) by 2−k−1N causes an error on SN no greater

than 2−1P k≥02−m(2 k₎ . Thus SN = NPk≥02−k−12−m(2 k₎ + EN, where |E_N| ≤ 2−1X k≥0 2−m(2k)+ .5q/(q − 1) X n even ≤N q−n/2− 2−m(1)q−1. But .5q/(q − 1)P n even ≤Nq−n/2≤ 2(q−1)q q−1 1−q−1 = .5q(q − 1)−2 and 2−1P k≥02−m(2 k₎ = 2−1 2−m(1)+P k≥12−m(2 k₎ = 2−1−m(1)+ 2−m(2). Therefore, |EN| ≤ .5q(q − 1)−2+ [2−1−ν2(q−1)+ 2−ν2(q 2₋₁₎ ] − 2−ν2(q−1)_q−1 < 3 8 + ₁ 4+ 1 8 − 0 = 6 8. (4.3) Dividing SN = NPk≥02−k−12−m(2 k₎

+ EN through by N yields the

theo-rem.

Remark 4.8. The bound on |E_N| given in (4.3) is better than 3/4 for most q’s.

5. The classical 1:2 odds revisited and why the choice of X is a representative choice

Definition. (Arithmetic density of a set of monic polynomials) The

arith-metic (or natural) density of a set A ⊂ N is defined, if it exists, as d(A) = limN →∞N−1AN, where AN is the number of natural integers in A and

≤ N . Note that d(A) = lim_{N →∞} 1 N

PN

n=1 χA(n)

1 , where χA is the

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arithmetic density of a set S of monic polynomials in Fq[X] as the limit

d(S) = limN →∞_N1 PNn=1 S(n)

qn , where S(n) is the number of elements of S

of degree n. Here qn _{stands for the number of monic polynomials in F} q[X]

of degree n, or of norm qn.

5.1. A heuristic for the expected odds in the classical case. We

start by recalling a fact which is familiar to people in the field of Lucas sequences: For “most" Lucas sequences Un= α

n_−βn

α−β , the density of primes

p with odd rank r(p) in (Un) is 1₃ (see [Ba1, Lag, Lax, Mo1] and [M-S]).

Here α and β are the roots of a quadratic X2− aX + b ∈ Z[X]. The rank r(p) is the least integer r ≥ 1 such that p Ur; it always exists if p6 2b. Although with the existing results in the area this fact could be turned into an explicit theorem, our intention is merely to provide an informal argument aimed at understanding. Why do primes first divide (Un) twice

more often at even indices?

Let us only consider Lucas sequences of the type a_a−1n−1 where a ∈ Z. Here r(p) is the order of a (mod p). And assume for simplicity that a is a positive integer. Let δa denote the density of primes p for which order

of a (mod p) is odd. From Theorem 3.1.3 of [Ba1] one can deduce that for any a ∈ A, δa = 1₃, where A = {a ∈ Z≥1; a = 2d· m, where d ≥

0 and m is a non-square odd integer}. Since the arithmetic density of A is one, we get that limN →∞_N1 PNa=1δa = 1₃. This is a proof, but it required the

knowledge of the existence and the value of the δa’s. Let us now compute

the average order of a (mod p) from a dual point of view. Instead of fixing an a and counting primes for which a has odd order, let us fix a prime p and evaluate the proportion α(p), in terms of arithmetic density, of positive in-tegers a having odd order (mod p). In computing α(p), we discard inin-tegers a divisible by p. For j ≥ 1, let Pj be the set {p prime ; ν2(p − 1) = j}. If

p ∈ Pj, then an integer a prime to p has odd order (mod p) if and only if a is in the unique subgroup of order p−1₂j of (Z/p)

∗_{. Hence α(p) =} 1

2j. So it is

never 1₃. But what about the average over all p’s? By the Dirichlet Density Theorem each Pj possesses a density δ(Pj) = ₂1j, so it seems reasonable1

to assign to each possible value of α(p) a weight equal to the “probability" that p be in Pj, i.e. δ(Pj). Then this average is

X j≥1 δ(Pj) · 1 2j = X j≥1 1 4j = 1 4 · 1 1 − 1/4 = 1 3 !

Remark. Had we considered the proportion β(p), in terms of Dirichlet

density, of primes q for which order of q (mod p) is odd, rather than the proportion α(p) of positive integers a, then the Dirichlet Density Theorem

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would have given β(p) = α(p) and therefore we would have obtained an expected probability of 1₃ again.

5.2. Expected odds in Fq[X]. Let P be a prime of Fq[X] of degree d. Let

R ∈ Fq[X] be of degree < d. Then A = {M ∈ Fq[X]; M is monic and M ≡

R (mod P )} has an arithmetic density. Indeed suppose n is any integer ≥ d. Then M of degree n belongs to A if and only if M = R + ΛP with Λ a monic polynomial of degree n − d. Hence d(A) = qn−d_qn = _{|P |}1 .

To emulate what we did in 5.1 let α(P ) be the proportion, in terms of arithmetic density, of monic polynomials M in Fq[X] for which the order

of M (mod P ) is odd. In evaluating α(P ) we discard those M divisible by P . Hence there are |P | − 1 disjoint sets of arithmetic density 1/|P | corresponding to the |P | − 1 classes in (Fq[X]/P )∗. Exactly ₂_{ν2(|P |−1)}|P |−1 have

odd order (mod P ). Hence α(P ) = 1/2ν2(|P |−1)_.

As noticed in Section 3 the function α(P ) is constant on Sq(k). Thus as

above we write that the average of the α(P ) as P varies through the primes of Fq[X] is (5.1) X k≥0 δ(Sq(k)) · α(P ∈ Sq(k)) = X k≥0 1 2k+1 · 1 2m(k) = X k≥0 δ(Oq(k)) = δq!

Thus the polynomial X, unlike the prime 2 of Sierpinski, behaves like a typical polynomial.

Remark. By the Dirichlet Density Theorem the set {Q ∈ Fq[X]; Q prime

and Q ≡ R (mod P )}, where R is a representative of a class in (Fq[X]/P )∗,

has a density = _{Φ(P )}1 = _qd1₋₁. Thus the sum in (5.1) can be interpreted as

the “probability" that, given P and Q ∈ Fq[X], the order of Q (mod P ) be

odd.

6. Density averages

Definition. In this Section we write δ(p, e) for the density δq, where q = pe.

Let p1 = 3, p2 = 5, . . . , pn, . . . denote the odd rational primes in their

natural increasing order. Since a density δ(pn, e) is associated to each prime

power pe_n, we define most naturally the average δ(−, e) of the densities δ(p, e) over all primes p by the limit limN →∞N−1PNn=1δ(pn, e) if it exists.

Similarly we define δ(p, −) to be limEE−1PEe=1δ(p, e), and the average

over all q0s, δ(−, −), as limNN−2PNn=1

PN

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Theorem 6.1. The average densities defined above all exist and are given,

for odd p, by the formulas δ(−, e) = ( ₇ 36, if e is odd, 2−ν2(e)_· 4 36, if e is even, δ(p, −) = 36−1[9 · 2−ν2(p−1)_{+ 14 · 2}−ν2(p2−1)_] and δ(−, −) = 5 2 63 = 4 +1₆ 36 .

Proof. We only prove the formula for δ(−, e), since the other cases can be computed in similar fashion. Note in passing that both limits

lim N N −1XN n=1 δ(pn, −) and lim E E −1XE e=1 δ(−, e)

turn out to also be equal to δ(−, −). So assume e to be a fixed integer ≥ 1. Let Pu = {p; ν2(p − 1) = u} and P+v = {p; ν2(p + 1) = v} for u and

v integers ≥ 1. By the Dirichlet Density Theorem for primes in arithmetic progressions, we have δ(Pu) = 2−u and δ(P₊v) = 2−v, since for example p ∈ Pu ⇐⇒ p ≡ 1 + 2u _{(mod 2}u+1_{) and ϕ(2}u+1_{) = 2}u_{. The constant}

values of δ(p, e) on each Pu_{, u ≥ 2, and on each P}v

+, v ≥ 2, are denoted

respectively by δu and δv₊. Since primes in arithmetic progressions have a natural density d equal to their Dirichlet density ([Pra], Chap. 5), we have d(Pu) = 2−u and d(P₊v) = 2−v.

Theorem 3.3 gives the values of δu and δ₊v for u ≥ 2 and v ≥ 2. We claim that the contribution of primes p ≡ 1 (mod 4) to δ(−, e) isP

u≥2δ(Pu) · δu.

For instance if e is odd, it isP

u≥22−u·3−12−u+1 = 181. And the contribution

of primes p ≡ −1 (mod 4) is P

v≥2δ(P+v)δ+v. For e odd, it is

P

v≥22−v ·

(4−1+ 3−12−1−v) = ₃₆5 . Hence, for e odd, δ(−, e) = ₃₆2 +₃₆5.

To prove the above claim, let MN = _N1 PNn=1δ(pn, e) be the average of

the δ(p, e)’s over the first N odd primes. First we show that lim inf MN ≥ l

and then that l ≥ lim sup MN, where l =Pu≥2δ(Pu)δu+

P

v≥2δ(P+v)δ+v.

Let Pu(N ) and P₊v(N ) be the number of indices n, 1 ≤ n ≤ N , such that pn ∈ Pu and respectively pn ∈ P+v. Since MN = Pu≥2

Pu_{(N )} N δu + P v≥2 Pv +(N ) N δ v

+≥ MN(U, V ), ∀U ≥ 2, V ≥ 2, where

MN(U, V ) = U X u=2 Pu_{(N )} N δ u₊ V X v=2 Pv +(N ) N δ v +,

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we get as N → ∞, lim inf MN ≥ PUu=2d(Pu)δu+

PV

v=2d(P+v)δv+, for any

U and any V ≥ 2. Letting U and V go to ∞, the last inequality yields lim inf MN ≥ l.

On the other hand for any U and V ≥ 2, MN = MN(U, V ) + 1 N X u>U Pu(N )δu+ X v>V P₊v(N )δv₊ ≤ M_N(U, V ) + 1 N X u>U Pu_{(N ) +} 1 N X v>V Pv +(N ).

Now limN _N1 Pu>UPu(N ) = d {p; p ≡ 1 (mod 2U +1)}

= 2−U and limN _N1 Pv>V P+v(N ) = d {p; p ≡ −1 (mod 2V +1)}

= 2−V. Hence, ∀ > 0, for any U, V and N large enough we have MN ≤ MN(U, V ) + . As

N → ∞, we get lim sup MN ≤ PUu=2δ(Pu)δu+

PV

v=2δ(P+v)δ+v + , which

as U, V → ∞ yields lim sup MN ≤ l + .

Example. The average over all primes p of the densities δp4 is only ₃₆1 . Remark. No average density δ(−, e), δ(p, −) or δ(−, −) is ever equal to any

δq, q a prime power, contrary to the 1/3 average, in the classical setting, of

the densities of the sets Tp= {` rational prime; order of p (mod `) is odd}

as p varies through rational primes.

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Christian Ballot

Département de Mathématiques, Université de Caen, Campus 2, 14032 Caen Cedex, France E-mail : ballot@math.unicaen.fr